In order for the convergence of our solver to be robust, we require that \(c_1\) 
is independent of \(h\) , \(p\) , \(\alpha \) , and \(\beta \) . For example, the slow
convergence of the Jacobi iteration on the Poisson problem is because the space
decomposition is not stable in the mesh parameter \(h\) . This may be addressed
by incorporating a global coarse space (on a coarser mesh with \(H > h\) ) into
the space decomposition (as in a two-level domain decomposition method, or a
multigrid method), which gives a stability constant independent of \(h\)  for the
Poisson equation. However, this space decomposition is still not stable in
\(p\)  for (REF ), nor is it stable in \(h\) , \(\alpha \) , \(\beta \)  for
() or (). Essentially, this is because for \(k=0\) 
eigenfunctions associated with small eigenvalues of the operator are smooth and
can be well-represented on a coarse grid, but this is not true for \(k \in \lbrace 1,2\rbrace \) . Building on work by Pavarino [1]}, Arnold, Falk &
Winther [2]}, [3]}, and Hiptmair [4]}, we now
discuss the space decompositions proposed by Schöberl and
Zaglmayr [5]}, [6]} that are robust in \(h\) , \(p\) , \(\alpha \) ,
and \(\beta \)  in numerical experiments.
